Abstract Musical Intervals: Group Theory for Composition and Analysis (original) (raw)
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A general pitch interval representation: Theory and applications
Journal of New Music Research, 1996
Pitch and pitch-intervals are most often represented-in the western traditioneither by the traditional pitch naming system and the relating pitch-interval names, or as pitch-classes and pitch-class intervals. In this paper we discuss the properties, relationships and limits of these two representations and propose a General Pitch Interval Representation (GPIR) in which the above two constitute specific instances. GPIR can be effectively used in systems that attempt to represent pitch structures of a wide variety of musical styles (from traditional tonal to contemporary atonal) and can easily be extended to other microtonal environments. Special emphasis will be given to the categorisation of intervals according to their frequency of occurrence within a scale. Two applications of the GPIR will be presented: a) in a system that transcribes melodies from an absolute pitch number notation to the traditional staff notation, and b) in a pattern-matching process that attempts to discover repetitions within a melody.
Mathematics and group theory in music
The purpose of this paper is to show through particular examples how group theory is used in music. The examples are chosen from the theoretical work and from the compositions of Olivier Messiaen (1908-1992), one of the most influential twentieth century composers and pedagogues. Messiaen consciously used mathematical concepts derived from symmetry and groups, in his teaching and in his compositions. Before dwelling on this, I will give a quick overview of the relation between mathematics and music. This will put the discussion on symmetry and group theory in music in a broader context and it will provide the reader of this handbook some background and some motivation for the subject. The relation between mathematics and music, during more than two millennia, was lively, widespread, and extremely enriching for both domains. This paper will appear in the Handbook of Group actions, vol. II (ed. L. Ji, A. Papadopoulos and S.-T. Yau), Higher Eucation Press and International Press.
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While small concept lattices are often represented by line diagrams to better understand their full structure, large diagrams may be too complex to do this. However, such a diagram may still be used to receive new ideas about the inherent structure of a concept lattice. This will be demonstrated for a certain family of formal contexts arising from mathematical musicology.
An Advance on A Theory for All Music
Perspectives of New Music 30/1: 158-183, 1992
just before the semi-formal exposition of the specifically musical part of the theory, that (p. 55) "further reduction of the basis [i.e., of the group of undefined predicates in A Theoy for All Music] is possible by adopting the approach of [Nelson] Goodman's The Structure of Appearance [1966]."
No Publication, 2023
This paper explores parsimonious voice leading transformations within pitch-class sets, from the perspective of three distinct voice leading states: static retention, increasing intervals, and decreasing intervals, all under the condition of preserving the invariance of pitch-class sets with prime form. It introduces concepts related to the number of voices, rules of voice leading, transformational outcomes in sets, and classification and relational theories in the normal form of pitch-class sets. The significance of this theory extends the practical applications of bridging, hooking invariance, and hybrid relational combinations in transformations from trichords to hexachords. Furthermore, it broadens the scope of post-tonal triadic invariance in Neo-Riemannian theory by applying these characteristics to the overall pitch-class sets space, ultimately forming a comprehensive theory of parsimonious voice leading and the invariance of pitch-class sets with prime form within this space.
On the Algebraic Properties of Intervals and Some Applications
Reliable Computing, 2001
The algebraic properties of interval vectors (boxes) are studied. Quasilinear spaces with group structure are studied. Some fundamental algebraic properties are developed, especially in relation to the quasidistributive law, leading to a generalization of the familiar theory of linear spaces. In particular, linear dependence and basis are defined. It is proved that a quasilinear space with group structure is a direct sum of a linear and a symmetric space. A detailed characterization of symmetric quasilinear spaces with group structure is found.
Interval Mathematics: Foundations, Algebraic Structures, and Applications
We begin by constructing the algebra of classical intervals and prove that it is a nondistributive abelian semiring. Next, we formalize the notion of interval dependency, along with discussing the algebras of two alternate theories of intervals: modal intervals, and constraint intervals. With a view to treating some problems of the present interval theories, we present an alternate theory of intervals, namely the "theory of optimizational intervals", and prove that it constitutes a rich S-field algebra, which extends the ordinary field of the reals, then we construct an optimizational complex interval algebra. Furthermore, we define an order on the set of interval numbers, then we present the proofs that it is a total order, compatible with the interval operations, dense, and weakly Archimedean. Finally, we prove that this order extends the usual order on the reals, Moore's partial order, and Kulisch's partial order on interval numbers. Keywords. Classical interval arithmetic; Machine interval arithmetic; Interval dependency; Constraint intervals; Modal intervals; Classical complex intervals; Optimizational intervals; Optimizational complex intervals; S-field algebra; Ordering interval numbers.